<p>We study a generalisation of the one-dimensional Rademacher random walk introduced in Bhattacharya and Volkov (ALEA Lat. Am. J. Probab. Math. Stat. 20(1):33–51, 2023) to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> (for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, the Rademacher random walk is always transient, as follows from Theorem&#xa0;8.8 in Engländer and Volkov (Coin Turning, Random Walks and Inhomogeneous Markov Chains, World Scientific and Volkov, Singapore, 2025)). This walk is defined as the sum of a sequence of independent steps, where each step goes in one of the four possible directions with equal probability, and the size of the <i>n</i>th step is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(a_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\{a_n\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is a given sequence of positive integers. We establish some general conditions under which the walk is recurrent or transient.</p>